Bisectors in tetrahedron

I've got such a problem. In tetrahedron we're considering the three bisectors of angles which are having common vertex. How to proof that if two of this bisectors are perpendicular that implicates that the third one must be also perpendicular to this two bisectors??

I've got such a problem. In tetrahedron we're considering the three bisectors of angles which are having common vertex. How to proof that if two of this bisectors are perpendicular that implicates that the third one must be also perpendicular to this two bisectors??

Take that vertex to be the origin, and let p, q, r be unit vectors along the three edges which meet at the origin. Then p+q is a vector along the bisector of the angle between p and q. Suppose we know that this bisector is perpendicular to the bisector of the angle between p and r. Then the scalar product (p+q).(p+r) is 0. Therefore p.q + p.r + q.r = –1. But that equation is symmetric in p, q and r. Consequently, the other two pairs of bisectors are also perpendicular to each other.

Sorry but I don't understand your proof. If you could explain a little bit more precisely

If p and q are unit vectors (or more generally vectors with the same length) then the points represented by the vectors 0, p, p+q and q form the vertices of a rhombus. The diagonals of a rhombus bisect the angles at the vertices. Thus the vector p+q bisects the angle between p and q.

The other thing I used in the proof is that the condition for two vectors to be perpendicular is that their scalar product ("dot product") should be zero.

Does that help to make things clearer? If not, you'll have to give a bit more detail about what it is that you don't understand.

small question

Originally Posted by Opalg

If p and q are unit vectors (or more generally vectors with the same length) then the points represented by the vectors 0, p, p+q and q form the vertices of a rhombus. The diagonals of a rhombus bisect the angles at the vertices. Thus the vector p+q bisects the angle between p and q.

The other thing I used in the proof is that the condition for two vectors to be perpendicular is that their scalar product ("dot product") should be zero.

Does that help to make things clearer? If not, you'll have to give a bit more detail about what it is that you don't understand.

hello, i am also interested in this proof. I understand the whole idea, but could you explain how "(p+q).(p+r) is 0" implies "p.q + p.r + q.r = –1" ?